Graph coloring problems are arguably among the most fundamental graph problems in parallel and distributed computing with numerous applications. In particular, in recent years the classical ($\Delta+1$)-coloring problem became a benchmark problem to study the impact of local computation for parallel and distributed algorithms. In this work, we study the parallel complexity of a generalization of the ($\Delta+1$)-coloring problem: the problem of (degree+1)-list coloring (${\mathsf{D1LC}}$), where each node has an input palette of acceptable colors, of size one more than its degree, and the objective is to find a proper coloring using these palettes. In a recent work, Halld\'orsson et al. (STOC'22) presented a randomized $O(\log^3\log n)$-rounds distributed algorithm for ${\mathsf{D1LC}}$ in the ${\mathsf{LOCAL}}$ model, matching for the first time the state-of-the art complexity for $(\Delta+1)$-coloring due to Chang et al. (SICOMP'20). In this paper, we obtain a similar connection for $\mathsf{D1LC}$ in the Massively Parallel Computation (${\mathsf{MPC}}$) model with sublinear local space: we present a randomized $O(\log\log\log n)$-round ${\mathsf{MPC}}$ algorithm for ${\mathsf{D1LC}}$, matching the state-of-the art ${\mathsf{MPC}}$ algorithm for the $(\Delta+1)$-coloring problem. We also show that our algorithm can be efficiently derandomized.

翻译：图着色问题可以说是并行和分布式计算中最基础的图问题之一，具有广泛的应用。特别是近年来，经典的($\Delta+1$)-着色问题已成为研究局部计算对并行和分布式算法影响的一个基准问题。在本文中，我们研究($\Delta+1$)-着色问题的一个推广——(度+1)-列表着色(${\mathsf{D1LC}}$)问题的并行复杂度，其中每个节点都有一个可接受颜色的输入调色板，其大小比该节点的度多一，目标是利用这些调色板找到一种正确的着色。在最近的一项工作中，Halld\'orsson等人（STOC'22）在${\mathsf{LOCAL}}$模型中提出了一种随机化$O(\log^3\log n)$轮的${\mathsf{D1LC}}$分布式算法，首次匹配了Chang等人（SICOMP'20）针对($\Delta+1$)-着色问题的最新复杂度。在本文中，我们在具有次线性局部空间的大规模并行计算（${\mathsf{MPC}}$）模型中为$\mathsf{D1LC}$获得了类似的联系：我们提出了一种随机化$O(\log\log\log n)$轮的${\mathsf{MPC}}$算法用于${\mathsf{D1LC}}$，匹配了针对($\Delta+1$)-着色问题的最先进${\mathsf{MPC}}$算法。我们还展示了我们的算法可以有效地去随机化。